Chapter 2: Problem 1
Find the sum of the series \(\frac{x^{2}}{1.2}-\frac{x^{3}}{2.3}+\frac{x^{4}}{3.4}-\ldots+(-1)^{n+1} \frac{x^{n+1}}{n(n+1)}+\ldots,|x|<1\)
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Prove that (i) \(\int_{1}^{2} \frac{d x}{(x+1) \sqrt{x^{2}-1}}=\frac{1}{\sqrt{3}}\). (ii) \(\int_{0}^{1} \frac{\mathrm{dx}}{(1+x)(2+x) \sqrt{x(1-x)}}=\pi\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{6}}\right)\).
Explain why each of the following integrals is improper. (a) \(\int_{1}^{\infty} x^{4} e^{-x^{4}} d x\) (b) \(\int_{0}^{\pi / 2} \sec x d x\) (c) \(\int_{0}^{2} \frac{x}{x^{2}-5 x+6} d x\) (d) \(\int_{-\infty}^{0} \frac{1}{x^{2}+5} d x\)
Prove that (i) \(\int_{1}^{2} \frac{d x}{x^{3}+3 x+1}<\frac{1}{5}\) (ii) \(3 \sqrt{23}<\int_{2}^{5} \sqrt{3 \mathrm{x}^{3}-1} \mathrm{dx}<10 \sqrt{15}-8 \sqrt{6} / 5\) (iii) \(2<\int_{0}^{4} \frac{d x}{1+\sin ^{2} x}<4\) (iv) \(\frac{\pi}{2}<\int_{0}^{\pi / 2} \frac{\mathrm{d} \theta}{\sqrt{1-\mathrm{k}^{2} \sin ^{2} \theta}}<\frac{\pi}{2 \sqrt{1-\mathrm{k}^{2}}}\left(0<\mathrm{k}^{2}<1\right)\).
If a is positive and \(\mathrm{I}=\int_{-1}^{1} \frac{\mathrm{dx}}{\sqrt{1-2 \mathrm{ax}+\mathrm{a}^{2}}}\) then show that \(\mathrm{I}=2 \mathrm{ifa}<1\) and \(\mathrm{I}=\frac{2}{\mathrm{a}}\) if \(\mathrm{a}>1\).
Which of following integrals are improper ? Why? (a) \(\int_{1}^{2} \frac{1}{2 x-1} \mathrm{dx}\) (b) \(\int_{0}^{1} \frac{1}{2 x-1} d x\) (c) \(\int_{-\infty}^{\infty} \frac{\sin x}{1+x^{2}} d x\) (d) \(\int_{1}^{2} \ln (x-1) d x\)
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